{"ID":2842889,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.09423","arxiv_id":"2511.09423","title":"Pseudo-Differential Operators and Generalized Random Fields over Tori","abstract":"Matérn covariance functions are ubiquitous in spatial statistics, valued for their interpretable parameters and well-understood sample path properties in Euclidean settings. This paper examines whether these desirable properties transfer to manifold domains through rigorous analysis of Matérn processes on tori using pseudo-differential operator theory. We establish that processes on $d$-dimensional tori require smoothness parameter $ν\u003e 3d/2$ to achieve regularity $C^{(ν-3d/2)^-}_{\\text{loc}}$, revealing a dimension-dependent threshold that contrasts with the Euclidean requirement of merely $ν\u003e 0$. Our proof employs the Cardona-Martínez theory of pseudo-differential operators, providing new analytical tools to the study of random fields over manifolds. We also introduce the canonical-Matérn process, a parameter family that achieves regularity $C^{(ν-3d/2+2)^-}_{\\text{loc}}$, gaining two orders of smoothness over standard Matérn processes.","short_abstract":"Matérn covariance functions are ubiquitous in spatial statistics, valued for their interpretable parameters and well-understood sample path properties in Euclidean settings. This paper examines whether these desirable properties transfer to manifold domains through rigorous analysis of Matérn processes on tori using ps...","url_abs":"https://arxiv.org/abs/2511.09423","url_pdf":"https://arxiv.org/pdf/2511.09423v1","authors":"[\"Nicolas Escobar-Velasquez\"]","published":"2025-11-12T15:37:23Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}